Neural Fokker-Planck equation

Neural Fokker-Planck equation

In this talk, we develop and analyze numerical methods for high dimensional Fokker-Planck equations by leveraging generative models from deep learning. Our starting point is a formulation of the Fokker-Planck equation as a system of ordinary differential equations (ODEs) on finite-dimensional parameter space with the parameters inherited from generative models such as normalizing flows. We call such ODEs neural parametric Fokker-Planck equation. The fact that the Fokker-Planck equation can be viewed as the L2-Wasserstein gradient flow of Kullback-Leibler (KL) divergence allows us to derive the ODEs as the constrained L2-Wasserstein gradient flow of KL divergence on the set of probability densities generated by neural networks. For numerical computation, we design a variational semi-implicit scheme for the time discretization of the proposed ODE. Such an algorithm is sampling-based, which can readily handle Fokker-Planck equations in higher dimensional spaces. Moreover, we also establish bounds for the asymptotic convergence analysis of the neural parametric Fokker-Planck equation as well as its error analysis for both the continuous and discrete (forward-Euler time discretization) versions. Several numerical examples are provided to illustrate the performance of the proposed algorithms and analysis.

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Wuchen Li

May 06, 2020
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